Wavelet Analysis of Light Curves
By Evan Pollock
Fourier Transform has been the standard tool for signal processing for quite some time. Recently a new tool has emerged as an alternative to Fourier analysis. The new tool is the wavelet transform. Wavelet transform has obvious advantages over the Fourier transform. It is able to analyze a signal with respect to frequency, amplitude and time. The ability to resolve a signal with respect to time is very valuable for signals that are not constant and whose components vary over the duration of the signal. The wavelet analysis is able to analyze the signal with respect to time by choosing many different intervals then applying the transform to that that interval. All the intervals have the same area but the demotions are changed to gain the appropriate frequency or time resolution. For the typical light curve the wavelet programs runs through roughly 60 different intervals. The entire original signal can then be reconstructed from the wavelet output. The Wavelet program was written by Torrence, C. and G. P. Compo and can be found at the following web address, http://paos.colorado.edu/research/wavelets/. This program will compute wavelet transform of a time series and the level of significance of the power spectrum. The program allows the users to select the mother wavelet to be used, the wave number, pad the data series, use red or white noise as a background spectrum, and select the level of significance for the significance test. For our purposes the Morlet mother wavelet was selected. It was selected because it is based on a Sin wave and a Gaussian curve, so it is simple. It also has the best frequency and time resolution at low wave numbers. Increasing the wave number increases the frequency
sensitivity, but also decreases the amount of the low frequency spectrum that is not influenced by wraparound error. The area that is wraparound free is defined as the area above the cone of influence (COI). The COI is plotted over the final output and changes as the wave number is changed. The wavelet transforms advantages are apparent but the complexity and number of parameters that affect the transform are a disadvantage. False positives will not occur but is possible to overlook significant signals if the parameters are not set correctly. To avoid this several steps have been taken. They are described below. To test the wavelet program synthetics light curves were made. The equation used to create the synthetic data is as follows: y=A*sin(findgen(300))+ sin(findgen(300)/20) + cos(findgen(300)/200)+ (randomu(seed,300)-.5)+ ((sin((findgen(300)/30)^2))) +A*sin(findgen(300)*3000) [A the is the amplitude factor for the high frequency signal] The first test contained a signal at a higher frequency than the Nyquist (frequency = 1/[2*dt]) limit, a high frequency within the testable range, one relatively medium and one relatively low frequency. A signal that changed frequency with time was also added to the signal. The period of this signal changed by at least a factor of ten over the duration of the spectrum. The results were very positive. In order for the high frequencies to be observable, not even significant, their amplitude needed to be three times that of the other frequencies. The signal above the Nyquist limit was not show in the plot except for a small area that was not significant. The next test was that was done was to see how the program reacted when data points were not present. 24 % of the data points from the synthetic data. The data gaps
were in three groups, a small 3% group, a medium 8% group, and large 13% group. This was done because the actual data is not sampled on a regular basis and some data sets have significant times where no data was collected. The wavelet program reacted favorably. Do to wraparound the “holes” still showed a signal present but it was not significant. Also the false signals tended to be either shifted up or down in period. The final observation was that removing these data points affected the COI. During the time when the data was removed the COI line was horizontal. None of these occurrences will negatively affect the final wavelet transform. Five different runs of the synthetic data can be seen in figure 1. After testing the actual data using different wave numbers a wave number of six was selected. It offers good frequency resolution and without sacrificing the range of frequencies to be tested. Wave numbers ranging from four to nine were tested, all of which produced a usable spectrum. This range of wave numbers was tested on actual data that ranged in cadence from two seconds to eight seconds. A wave number of six produced the best results at each of the cadences tested. Raising the wave number above nine lowers the height of the observable peaks, substantially reducing the significant areas. A wave number less than four does not have enough resolution to make a consistent power spectrum at high frequency. The wavelet transform will only work when the time series contains a 2 n number of elements. To accommodate data sets that do not contain this number of elements the program will pad the data series with zeros so it will contain the appropriate number of elements. This padding makes the program more efficient and thus quicker, as well as reducing the amount of wraparound error. Currently there is no lower limit on the
padding size, so a sequence containing 255 elements would be padded to 256. This is not optimal. When such a small padding occurs, wraparound errors should be very large. White noise was chosen and the background spectrum. White noise was chosen because most low frequency oscillations were removed. Testing red versus white noise, on both the synthetic data and real light curves, showed no significant differences. The keyword, LAG, if set to zero returns a white noise spectrum. If it is any other number a red noise background spectrum is used. The program requires an input of the time between data points, dt. Since our data does not have one consistent sampling rate we are given two options. First is to use the average sampling rate. This found by dividing the duration of the sequence by the number of images. The second option is to use the shortest time between two images. The first option is very reasonable but will show less high frequency data than the second option. However, dt is used to calculate levels of significance, COI, FFT, the wave number array, and in the Morlet mother wavelet function, making the second option riskier. After looking the data using the second option would only increase the high frequency range by a period of about 1.2. This small increase does not seem to warrant the manipulation of all the other parameters of the program. The wavelet analysis will give data that we then can compare to the DCDFT data for the same light curve. The wavelet output will give the duration of the oscillation. The DCDFT peaks can then be determined to be constant throughout the signal or just a quick oscillation. The wavelet data will show which type of oscillation, sustained or quick, is more prevalent in our data sequences.